Solving modulus equality for $x$.

Question

The given equation is: $$|x - |4-x|| -2x = 4.$$ (Here $|x|$ means the absolute value of $x [\text{abs}(x)])$ Please help me to solve the equation for $x$.

Answer 1

$|4-x|$ is either $4-x$ or $x-4$. In the first case, we want to solve $|x-(4-x)|-2x=4$, in the other $|x-(x-4)|-2x=4$. In the latter case, $|x-(x-4)|$ is simply $4$, and $4-2x=4$ has the only solution $x=0$. In the first case again, $|x-(4-x)|$ is either $2x-4$ or $4-2x$, so we want to solve $-4=4$ (nope) or $4-4x=4$ (leads to $x=0$). So we only find $x=0$ as candidate solution and readily verify that it actually solves the original equation.